ABOUT MAGNETIC ADS BLACK HOLES
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Published for SISSA by Springer
Received: November 25, 2020
Accepted: January 16, 2021
Published: March 5, 2021
About magnetic AdS black holes
JHEP03(2021)068
Brett McInnes
Department of Mathematics, National University of Singapore,
10 Lower Kent Ridge Road, Singapore, Republic of Singapore
E-mail: matmcinn@nus.edu.sg
Abstract: There has recently been a strong revival of interest in quasi-extremal mag-
netically charged black holes. In the asymptotically flat case, it is possible to choose the
magnetic charge of such an object in such a manner that the black hole is surrounded by
a corona in which electroweak symmetry is restored on macroscopic scales, a result of very
considerable interest. We argue that holographic duality indicates that the asymptotically
AdS analogues of these black holes have several interesting properties: the dual theory
is only physical if the black hole is required to rotate; in the rotating case, the magnetic
field at the poles does not attain its maximum on the event horizon, but rather somewhat
outside it; the magnetic field at the equator is not a monotonically decreasing function of
the magnetic charge; the electric fields induced by the rotation, while smaller than their
magnetic counterparts, are by no means negligible; the maximal electric field often occurs
neither at the poles nor at the equator; and so on. Most importantly, in the magnetically
charged case it is possible to avoid the superradiant instability to which neutral AdS-Kerr
black holes are subject; but the need to avoid this instability imposes upper bounds on the
magnetic and electric fields. In some circumstances, therefore, the corona may not exist in
the asymptotically AdS case.
Keywords: AdS-CFT Correspondence, Black Holes, Black Holes in String Theory
ArXiv ePrint: 2011.07700
Open Access, c The Authors.
https://doi.org/10.1007/JHEP03(2021)068
Article funded by SCOAP3 .Contents
1 Magnetic black holes in AdS must spin 1
2 Magnetic AdS4 -Kerr-Newman black holes 4
3 Avoiding superradiance: a bound on magnetic charge 7
4 Where are the fields maximal? 9
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5 Bounds on the fields 11
6 The chemical potential 16
7 Conclusion 18
1 Magnetic black holes in AdS must spin
It has long been assumed that the electric charge on astrophysical black holes must be
negligible. Recently, however, this belief has been challenged, both for specific observational
reasons connected with mergers of black holes [1], and for more general reasons based on the
sensitive dependence of the radius of the innermost stable circular orbit on the charge [2].
The case of rotating charged black holes is particularly interesting, since the usual
elementary argument in favour of electrical neutrality can apparently be circumvented in
this case. Nevertheless, a very detailed analysis of this question [3] suggests that even these
black holes are probably electrically neutralized (though a marginal loophole, in the case
where a magnetosphere is present, remains to be fully closed). For clarity, we will therefore
assume henceforth that the electric charge of the black holes to be considered here is zero.
The case of magnetically charged black holes [4] has also begun recently to attract
renewed attention. Because magnetic monopoles are rare and probably extremely mas-
sive, magnetic black holes differ from their electric counterparts: they are not readily
neutralized; if they can form at all, they will persist. They are therefore potentially of
observational interest. For example, magnetic charge on black holes can affect the merger
time of binaries [5]; again, it is possible, in certain theories, to constrain magnetic charges
by studying the “shadows” around supermassive black holes [6]. More dramatically, it has
been suggested [7–9] that such objects may give rise to enormous magnetic fields, which
can restore the full electroweak gauge group in a corona adjacent to the event horizon. All
this has led recently to much interest in the phenomenology of these objects: see [10, 11]
for comprehensive treatments of these developments.
From a theoretical point of view, magnetically charged black holes may lead to a better
understanding of the electroweak interaction in a very extreme environment, and similarly
of quantum chromodynamics in the form it takes in the presence of extremely strong
–1–magnetic fields. (This in itself is a subject which has attracted much interest lately [12, 13].)
One might hope in this way to gain some insight into theories beyond the Standard Model.
It is therefore interesting to embed such black holes in string theory. The natural way
to begin such a programme is to consider the asymptotically anti-de Sitter versions of these
objects.
Magnetically charged asymptotically AdS4 black holes were discussed briefly in [7],
where in particular it is pointed out that, by manipulating the magnetic field on the
conformal boundary of AdS4 , one can arrange for a pair of magnetic black holes to be
located at desired positions in the bulk spacetime, with magnetic field lines threading both
JHEP03(2021)068
black holes. This is useful for the study of four-dimensional traversable wormholes, but
these objects are clearly of more general interest.
This raises the obvious question: can we exploit the AdS/CFT duality [14–16] to
constrain magnetically charged asymptotically AdS4 black holes?1 At the very least, we
should be able to show that these black holes are not dual to some system that might be
completely unphysical.
In [7–9] it is shown that, due to its Hawking radiation, the magnetic black hole will
evolve towards being nearly extremal; precisely because of the existence of the corona, this
happens very rapidly, and so near-extremality will be the generic state of the black hole. 2
Such a black hole has a very low Hawking temperature, and hence the dual matter in
the asymptotically AdS case is (relatively) cold. Furthermore, the magnetic charge on the
black hole induces a non-zero magnetic field B ∞ at infinity, so the dual matter is subjected
to a substantial magnetic field.
Low-temperature strongly coupled (quark) matter can and probably does exist [23], in
the cores of certain neutron stars; furthermore, a particular variety of neutron stars, the
magnetars [24], often contain matter exposed to very large magnetic fields. Thus it appears
that asymptotically AdS4 magnetically charged extremal black holes are holographically
dual to matter that is just a more extreme version of a form of matter which probably does
actually exist, in the cores of certain neutron stars. Thus, it begins to seem that extremal
magnetic AdS black holes are dual to a form of matter that is at least not obviously
unphysical.
However, an important parameter describing any form of strongly coupled matter is
the chemical potential µ [25] representing, in this case, the matter density. Cold matter of
this kind always has a large chemical potential [26]. In the case we are considering here,
we can interpret “large” to mean that it is large relative to the maximal value of the other
relevant quantity with the same dimension, namely the square root of the magnetic field,
B. In a magnetar, the field is huge by ordinary standards, typically around 1014 gauss;
1
For the details of the AdS/CFT correspondence for a magnetically charged four-dimensional bulk black
hole, see for example [17]; for the general AdS/CFT correspondence for a four-dimensional bulk, see [18, 19].
2
See [20] for the details of the effects of Hawking radiation in such cases; the evolution is not completely
straightforward and can proceed in unexpected ways. For example, Hawking radiation can also cause a
rotating black hole to evolve initially towards extremality [21, 22], though Censorship is never violated.
Note also that astrophysical magnetic black holes can be temporarily perturbed away from extremality by
their environments [10].
–2–√
but this means B ≈ 2.6 MeV, compared to µ which is estimated [26] to be in the range
of 2 to 6 times nuclear matter density, that is, from 600 MeV to well over 1000 MeV. While
we are not suggesting that the dual matter in our case closely resembles the quark matter
which may exist in the core of a magnetar, this discussion gives us a guide as to our general
expectations: the chemical potential induced at infinity by a magnetic AdS extremal black
hole should be at least of the order of magnitude of the square root of the magnetic field
at infinity, probably larger.
This is a serious problem here, because the time component of the electromagnetic
one-form at infinity for a magnetic AdS-Reissner-Nordström black hole vanishes, and the
chemical potential is represented holographically by this component; 3 so the matter dual to
JHEP03(2021)068
such a black hole has zero chemical potential. It seems, then, that magnetic AdS-Reissner-
Nordström extremal black holes are dual, that is, equivalent, to an entirely unphysical
system (strongly coupled matter with both low temperature and negligible matter density)
on the boundary.
We will show that the way to resolve this problem is to make the black hole rotate: that
is, to consider magnetic AdS4 -Kerr-Newman extremal black holes instead of their AdS4 -
Reissner-Nordström counterparts. This induces a non-zero angular momentum density at
infinity, but also, more importantly here, a constant but non-zero timelike component in
the electromagnetic one-form at infinity; thus we have a non-zero chemical potential there.
(It also induces an electric field around the black hole, though not at infinity.) Note that
other forms of strongly coupled matter in nature, such as those generated by a generic
collision of heavy ions, also typically have high angular momentum densities [28, 29]; so,
again, we are considering a more extreme version of a form of matter which does exist. In
the case at hand, in view of our discussion above, we will endeavour to use this effect to
√
raise µ to at least the same order of magnitude as B ∞ .
We stress that we are not attempting to construct a realistic holographic model of any
form of quark matter here. Our objective, rather, is to show that magnetic AdS extremal
black holes need not be dual to a flagrantly unphysical system.
Even with this understanding, it is by no means clear that this stratagem can be made
to work. It is known [30] that all near-extremal AdS4 -Kerr black holes are unstable against
the well-known superradiant instability [31]. One might suppose that this is also true in the
near-extremal AdS4 -Kerr-Newman case. If that were so, then adding angular momentum
would not solve our original problem.
One objective of this work is to show that these two problems “solve each other”.
Making the purely magnetic extremal black hole rotate can induce a (reasonably large)
chemical potential in the theory at infinity, provided that the angular momentum is suffi-
ciently large; in turn, however, a sufficiently large magnetic charge stabilizes the rotating
black hole against superradiance. We will give a concrete example of a black hole which
embodies both of these virtues.
These results are of course welcome, but clearly they impose potentially interest-
ing bounds on the magnetic charge. Exploring these bounds is another objective of the
present work.
3
See [27] for a recent discussion of this, in the electric case.
–3–We are primarily interested in the maximal possible field values (both magnetic and
electric) around the black hole. In the non-rotating case, it is clear that the maximal
magnetic field occurs on the event horizon, and that, in the extremal case of interest to us
here, it is there inversely proportional to the magnetic charge. Leaving aside charge quan-
tization [32], then, one can attain arbitrarily large field values simply by taking the charge
to be sufficiently small; this was exploited in [8] to demonstrate the possible existence of
coronae around magnetic black holes. In this work, we will investigate the maximal values
of the fields in the near-extremal, rotating case.
We find that there are bounds on the maximal magnetic and electric fields attainable
around a rotating purely magnetic AdS4 extremal black hole, imposed by the requirement
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that the system be stable against superradiance. In order to explore these bounds, we need
to answer two questions:
[1] Since the black hole has to rotate, it is no longer spherically symmetric, and so nor is
the magnetic field (or the induced electric field). Thus we need to determine precisely
where the magnetic and electric fields attain their maxima as functions of position
(outside and on the event horizon).
[2] Avoiding superradiance imposes a lower bound on the magnetic charge; this lower
bound depends on the angular momentum parameter a. We need to understand how
the magnetic and electric fields vary with the charge, and to maximize them as a
function of the charge (and therefore of a).
Combining [1] and [2], we will be able to determine the maximal possible fields for
a black hole of this type, given its angular momentum parameter (which is determined
holographically by the chemical potential of the field theory at infinity).
These bounds are fixed by the angular momentum parameter of the black hole, by
the asymptotic AdS curvature scale L, and by the AdS gravitational coupling, or, dually,
by the chemical potential, the magnetic field at infinity, and the number of M2-branes in
the field theory at infinity. This means that, in some cases, even the maximal magnetic
and electric fields may be too small to restore the electroweak symmetry: there may be no
corona.
We begin with a brief discussion of the relevant properties of magnetic AdS 4 -Kerr-
Newman black holes.
2 Magnetic AdS4 -Kerr-Newman black holes
When the event horizon is topologically spherical, the four-dimensional magnetic AdS-
Kerr-Newman metric [33] has the form
" #2
∆r a ρ2 2 ρ2 2
g(AdSa,M,P,L ) = − 2 dt − sin2 θ dφ + dr + dθ (2.1)
ρ Ξ ∆r ∆θ
" #2
sin2 θ ∆θ r 2 + a2
+ 2
a dt − dφ ,
ρ Ξ
–4–where
ρ2 = r2 + a2 cos2 θ,
!
2 r2
2 P2
∆r = (r + a ) 1 + 2 − 2M r + ,
L 4π
a2
∆θ = 1 −cos2 θ,
L2
a2
Ξ = 1 − 2. (2.2)
L
The geometry is characterised by four parameters, all with dimensions of length: L is
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the asymptotic AdS4 length scale, a is the angular momentum per unit physical mass,
and M and P are geometric parameters related [35, 36] to the physical mass M and the
(dimensionless) physical magnetic charge P by
M = M/(`2G Ξ2 ), P = P/(`G Ξ), (2.3)
where `G is the gravitational length scale in the bulk.
The parameters a and L are related in an interesting way. If we allow a > L, then
the signature of the metric becomes direction-dependent, changing from (−, +, +, +) to
(−, +, −, −) for sufficiently small values of θ. Thus the r coordinate becomes timelike
(in the “mostly minus” signature convention) in some directions, and conformal infinity
in those directions is spacelike, as in de Sitter spacetime. While this is interesting, such
exotic objects are beyond our scope here, and we will impose the condition
a < L. (2.4)
Once this is agreed, Cosmic Censorship prohibits values of a arbitrarily close to L if the
physical mass and charge of the black hole are fixed: the upper bound on a/L is some
number strictly smaller than unity, with a precise value determined by those parameters. 4
The electromagnetic potential form here is given (see [33, 34, 37]) by
! !
P cos θ r2 + a2
aP cos θ aP P
A= − 2
+ 2 + a2
dt +
2
− dφ, (2.5)
4π`G ρ 4π`G rH 4π`G ρ 4π`G
where the constant terms are fixed by the fact that, in the Euclidean version of the ge-
ometry, ∂t and ∂φ vanish at the points corresponding to the poles of the Euclidean “event
horizon”; hence the potential one-form, if it is to be regular, must vanish at those points
when evaluated on those vectors, and this requires the inclusion of these constant terms.
Here rH is the value of the radial coordinate at the event horizon, which is of course related
to the other parameters by solving the equation
!
2 2 r2 P2
∆r (rH ) = (rH + a ) 1 + H2 − 2M rH + = 0 (2.6)
L 4π
for its largest root.
4
To avoid confusion later, however, we should stress here that, throughout our discussions, we do not
have any motivation or justification for fixing the mass. In fact, we are primarily interested in quasi-extremal
black holes; we will vary the angular momentum and charge independently, and take it that the mass is
adjusted so that extremality is maintained.
–5–Using the orthonormal basis forms with respect to which the fields are radial, one can
use (2.5) (and (2.3)) to evaluate the magnitude of the magnetic field outside the event
horizon [33]:
Ξ P r2 − a2 cos2 θ
B = . (2.7)
4π (r2 + a2 cos2 θ)2
We note in passing that the presence of the factor Ξ here can be understood as follows:
2 2
integrating the magnetic field against the element of area in this case, r +a Ξ sin θ dθ dφ
(for any surface defined by a fixed r), one obtains P, in agreement with Gauss’ law, only
because of the factor of Ξ on the right side of (2.7). This factor plays a crucial role in our
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subsequent analysis.
Despite the complete absence of electric charge, the rotation of the magnetically
charged black hole induces a non-zero electric field, with magnitude given (again by us-
ing (2.5) and (2.3)) by
−Ξ Pra cos θ
E = . (2.8)
2π (r2 + a2 cos2 θ)2
This field (which, far from the black hole, falls off in the typical manner of a dipole,
that is, as 1/r3 ) will attract or repel nearby electrically charged objects, but of course it
cannot be neutralized in that manner and is a permanent feature of the black hole if the
magnetic charge does not change. It is a complicated function of the parameters, and will
be considered in detail, below.
In order to study the holography of these black holes, we need to understand the
electromagnetic fields at conformal infinity. They are obtained by using the same conformal
factor5 as one uses to define the boundary metric. The result is that there is no electric
field at infinity, but there is a magnetic field at infinity:6
ΞP
B∞ = . (2.9)
4πL2
Notice that it depends not only on P but also on a, through Ξ.
The chemical potential of the strongly coupled matter at infinity is proportional to the
timelike component of the electromagnetic one-form at infinity. It is
3Ξ aP
µ = 2 + a2 .
(2.10)
4π rH
As we foresaw, this crucial quantity can only be non-zero in the magnetic case if a 6= 0:
the black hole must spin if the matter at infinity is to be even approximately physical. We
regard the chemical potential as being “given”, though only in the sense that it should
not be excessively small — as explained earlier, we are not attempting to set up an even
approximately accurate model of realistic quark matter here. As µ is induced by the black
5
The electromagnetic field tensor is conformally invariant: therefore its components relative to an or-
thonormal basis are not. Thus one computes the magnetic field at infinity.
6
It is customary in studies of collisions of heavy ions [38] to focus on a two-dimensional section through the
system, the reaction plane or “x − z plane”. The magnetic field vector is (to a good approximation) parallel
to the y-axis perpendicular to the reaction plane, so that only its magnitude plays a role. For magnetic
asymptotically AdS black holes, it is useful to think of the magnetic field at infinity in a similar way.
–6–hole angular momentum, we can regard the black hole parameter a as a proxy for it; we
adjust a so as to obtain the desired value or range of values for µ. Technically, this means
that we must endeavour to express our bounds in terms of quantities determined by a.
The Hawking temperature of the black hole is given [33] by
a2 + (P 2 /4π)
rH 1 + a2 /L2 + 2 /L2
3rH − 2
rH
T = 2) . (2.11)
4π(a2 + rH
In order to be more specific, we need to be able to express rH in terms of the other
parameters. Unfortunately, equation (2.6) being a quartic equation, this is difficult to do
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in general.
Fortunately, the problem is much simpler in the extremal case in which we are in-
terested here, because the condition of extremality is equivalent to setting the Hawking
temperature equal to zero. The result, from equation (2.11), is
a2 + (P 2 /4π)
1 + a2 /L2 + 3rH
2
/L2 − 2 = 0, (2.12)
rH
which means that
s
2
L2 a2 a2 12 P2
2
rH = −1 − 2 + 1+ 2 + 2 a2 + (2.13)
6 L L L 4π
in the extremal case. Using (2.3), we have finally
v !
`2 (1 − [a2 /L2 ])2 P 2
2
L2 a2 u
u
2 a2 12
rH = −1 − 2 + t 1 + 2 + 2 a2 + G . (2.14)
6 L L L 4π
This concludes our discussion of the general properties of magnetic AdS4 black holes.
We now turn to the most pressing question for these objects, their stability against super-
radiance.
3 Avoiding superradiance: a bound on magnetic charge
It was observed in [30] that AdS4 -Kerr black holes are at risk of an instability due to
superradiance [31]; a sufficient condition for avoiding this is
2
rH
aL 1 + L2
ξ ≡ 2 + a2 ≤ 1. (3.1)
rH
There is strong reason [39] to think that this condition is also necessary.7
7
Here, by “superradiance”, we mean the usual effect due to the existence of an ergosphere. There is
another form of black hole “superradiance” (see for example [40, 41]) due to the effects of charge. In the
magnetic case, however, this is most unlikely to be significant, since it would involve magnetically charged
scalar fields of very low (or zero) mass.
–7–It was pointed out in [39] that extremal (and therefore, far worse, sufficiently near-
extremal) AdS4 -Kerr black holes always violate (3.1). If this were also true of the near-
extremal magnetic AdS4 -Kerr-Newman black holes with which we are concerned here, then
we could not assume that such a black hole can be quasi-stationary, since it would shed its
angular momentum to superradiant modes. Thus, since we have argued that holography
indicates that these black holes must rotate, it is essential to settle this question.
Using (2.14), we have in the extremal case
r !
2
a2 a2
a
L 5− L2
+ 1+ L2
+ 12
L2
a2 + `2G (1 − [a2 /L2 ])2 P 2 /4π
JHEP03(2021)068
ξ = r 2 . (3.2)
5a2 a2
−1 − L2
+ 1+ L2
+ 12
L2
a2 + `2G (1 − [a2 /L2 ])2 P 2 /4π
It is important to note that this expression has a singular limit as P → 0: in that limit, this
function exceeds unity, for all values of the other parameters (including a): this confirms
that all extremal AdS4 -Kerr black holes are unstable to superradiance.
However, if all of the parameters except P are fixed, and ξ is regarded as a function
of P, then one finds that it is a decreasing function; and in all cases, it drops below unity
for sufficiently large P. Let us be more explicit about this, using a concrete example.
In order to ensure that the geometry can be self-consistently be treated as (approx-
imately) classical, we need the AdS curvature scale L to be significantly larger than the
Planck length; this is a familiar requirement in studies of the AdS/CFT correspondence.
To be definite, we set L = 100 in Planck units;8 other choices can of course be made, but
that does not affect our general conclusions. Guided by (2.4), we set a = 50. These will
be our standard choices henceforth, when we need a concrete example.
For this example, the graph of ξ(P) is as shown in figure 1. With these choices of L
and a, one finds that ξ ≤ 1 for all values of P exceeding approximately 501.3.
In short, the presence of magnetic charge allows us to avoid superradiance, provided
that the magnetic charge is sufficiently large.
The minimal permissible magnetic charge is found by setting ξ = 1. After some
simplification, one finds that, provided that a is strictly greater than zero (to avoid the
singular situation when P = 0), one can avoid a superradiant instability provided that the
magnetic charge on the extremal black hole satisfies
√
2 πaL
P ≥ Pmin = . (3.3)
`G 1 − La
It turns out that, if one fixes all other parameters and regards the magnetic field as
a function of P, then it is not a monotonic function; so the consequences of (3.3) for the
magnetic and electric fields are not immediately clear. We will return to that after we
determine where the maximal fields are to be found.
8
When discussing explicit examples, we work with AdS Planck units: that is, we set `G = 1.
–8–JHEP03(2021)068
Figure 1. ξ(P) with L = 100, a = 50. (Units are Planck units.)
4 Where are the fields maximal?
We are interested in the case in which the magnitude of the magnetic and electric fields
on and outside the event horizon is maximal for a given magnetic charge and angular
momentum parameter.
Let us begin with the magnetic field, given by equation (2.7). It is clear from inspection
of equation (2.7) that B(r, θ) is, for any fixed r, an increasing function of θ. On the event
horizon, then, the maximum occurs at the equator. However, we have to be careful: in the
non-rotating case, it is obvious that B(r, θ) takes its maximum, on the domain r ≥ rH , on
the event horizon; but it is not clear that this is so in the rotating case. Similar remarks
apply to the electric field. Let us consider the radial variation of the magnetic field.
One can show that, for any fixed value of θ, B(r, θ) has a local maximum at
√
r = r∗ = 3a cos(θ). (4.1)
Clearly this means that, on and near the equator, the local maximum does not occur on
or outside the event horizon. In those cases, then, the maximum on the domain r ≥ rH is
on the horizon, just as it is in the non-rotating case; and therefore the global maximum is
at the equator.
Near the poles, however, it may or may not be the case that the local maximum occurs
inside the event horizon. Consider the pole itself, θ = 0. Then the local maximum is inside
–9–√
the event horizon only if rH > 3a. Since rH is an increasing and unbounded function
of P for fixed a, we see that this will indeed be true for very large values of P: in those
cases, the maximal value of the magnetic field does occur on the horizon even at the poles.
√
But for smaller values of P, those such that rH < 3a, this need not be the case. Of
course, the only values of P of interest to us here are those satisfying the inequality (3.3),
so the question is whether there are values of the parameters such that (3.3) is satisfied
√
and yet rH < 3a.
The answer is that this is indeed possible; this is most clearly seen with the aid of a
concrete example, and we will proceed to that shortly. Meanwhile, notice that, if we are
at r = r∗ , we can always increase the field by increasing θ. However, if we do that, then
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from (4.1) we see that r∗ will decrease until eventually we are on the event horizon for some
value of θ < π/2. We can always increase the field still further by moving along the event
horizon to the equator. It follows that the global maximum of the field, on and outside the
event horizon, lies on the equator of the event horizon.
We can see how all this works by considering again our standard example, with L = 100
and a = 50 in Planck units.
In order to avoid superradiance, we need to take, as explained above, the magnetic
charge to be at least 501.3; let us take it to have that value. (As we will see later, this
choice of the charge maximizes the magnetic field, all else being equal; we stress again
that this is not obvious, because the field is not a monotonic function of the charge under
these conditions.) For an extremal black hole with these parameters, one finds from (2.14)
that rH ≈ 70.7.
Let us now see what happens at the poles, θ = 0. The graph of B (outside and on the
event horizon), which is now a function of r only, is shown as figure 2.
We see that the field does in this case attain a local maximum, as a function of r,
well outside the horizon (and therefore, incidentally, outside the ergosphere); it only then
decreases towards zero as r increases. This shows the need for caution in the rotating case.
However, as we explained earlier in the general case, this local maximum, and all of
the other local maxima of B as a function of r at intermediate values of θ, is always smaller
than the maximal value at the equator, which does occur exactly on the event horizon. In
the example we are considering here, this can be seen from figure 3: the maximal values
at the equator are far larger than the maximum seen in figure 2.
Unlike the magnetic field, the electric field does always attain its maximum at the
event horizon,9 for each fixed value of θ. However, on the event horizon, it does not always
attain its maximum at either the equator or the poles. One finds that the maximum is
√
attained at the poles when a/rH ≤ 1/ 3; otherwise, however, the maximum is attained at
an intermediate angle θ = θ+ , 0 ≤ θ+ < π/2 (and at the symmetrically located points in
the other hemisphere), where
rH
+
θ = arccos √ . (4.2)
3a
For the parameter values used above, one finds θ+ ≈ 0.62: see figure 4.
9
To be precise: as a function of r, the electric field has a local maximum for some fixed values of θ.
However, these local maxima always lie inside the event horizon.
– 10 –JHEP03(2021)068
Figure 2. The polar magnetic field Bp , as a function of the radial coordinate outside the event
horizon, for an extremal AdS4 -Kerr-Newman magnetic black hole with L = 100, a = 50, P = 501.3.
(Units are Planck units.)
With these parameter choices, the electric field at θ = θ+ ≈ 0.62 actually takes a larger
value (≈ 0.0039 in Planck units) than the magnetic field there (≈ 0.0022; this is the value
on the event horizon, where it turns out that the magnetic field takes its maximal value as
a function of r for this particular value of θ).
In short, the strongest field overall is the magnetic field at the equator, on the event
horizon. As we move along the event horizon, away from the equator towards either
pole, the magnetic field becomes weaker, but this is partially compensated by the induced
electric field, which is distributed in a complex manner that depends on the parameter
values (though it is always zero on the equator).
5 Bounds on the fields
The magnetic field on the event horizon is, from (2.7),
1 − [a2 /L2 ] P rH
2 − a2 cos2 θ
B(r = rH ) = 2 . (5.1)
2 + a2 cos2 θ
4π rH
– 11 –JHEP03(2021)068
Figure 3. The equatorial magnetic field, as a function of the radial coordinate outside the event
horizon, for an extremal AdS4 -Kerr-Newman magnetic black hole with L = 100, a = 50, P = 501.3.
(Units are Planck units.)
We saw earlier that the global maximum for the magnetic field, for fixed P and a,
occurs on the horizon, at the equator; so this maximum is
1 − [a2 /L2 ] P
+
B = 2 . (5.2)
4πrH
That is, B + is the maximal magnetic field, when the latter is regarded as a function of
position (not of the parameters a and P).
From equation (2.14) (and using (2.3)) we have now, in the extremal case,
3 1 − [a2 /L2 ] P
+
B = r 2
!. (5.3)
a2 a2
2πL2 −1 − L2
+ 1+ L2
+ 12
L2
a2 + `2G (1 − [a2 /L2 ])2 P 2 /4π
As an application of these explicit formulae, we now ask: how does the maximal
magnetic field vary with the magnetic charge?
Let us recall first the answers to this question for extremal asymptotically flat four-
dimensional magnetic black holes, and for extremal asymptotically AdS4 but non-rotating
magnetic black holes.
– 12 –JHEP03(2021)068
Figure 4. The electric field on the event horizon, as a function of θ, for an extremal AdS4 -Kerr-
Newman magnetic black hole with L = 100, a = 50, P = 501.3. (Units are Planck units.)
√
In the asymptotically flat, non-rotating case, one has simply rH = P`G / 4π; so the
area of the event horizon becomes smaller as the magnetic charge is reduced while main-
taining extremality.10 In that case the maximal magnetic field at the event horizon is just
+ 1
BAF = , (5.4)
P`2G
that is, it is large for small magnetic charge, as is stressed in [8]. (This is reminiscent of the
fact that the spacetime curvature near the horizon of a Schwarzschild black hole is largest
for black holes with the smallest masses.) Indeed, leaving aside the constraint imposed
by charge quantization, this means that the magnetic field at the event horizon can be
made arbitrarily large by choosing a sufficiently small P. Thus it is possible to attain such
large magnetic fields that the electroweak symmetry can be restored in a corona around
the event horizon.
This is still true even in the asymptotically AdS4 , non-rotating case: setting a = 0
in (5.3), we have
3P 1 3P
B + (a = 0) = = + + O(P 3 ), (5.5)
P`2G 4πL2
q
2πL2 −1 + 1+ 3`2G P 2 /πL2
which is of course unbounded above as P is reduced.
10
Of course, this entails a corresponding reduction of the mass, so this does not contradict the familiar
dictum that increasing the charge on a black hole of fixed mass reduces the entropy.
– 13 –JHEP03(2021)068
Figure 5. The magnetic field, evaluated on the event horizon at the equator, as a function of the
physical charge, for an extremal AdS4 -Kerr-Newman magnetic black hole with L = 100, a = 50.
(Units are Planck units.)
However, the analogous statement is false when the magnetic AdS4 black hole rotates,
no matter how small the angular momentum may be: from equation (5.3) we have
a2
3 1− L2
P
B+ = q + O(P 3 ); (5.6)
a2 a2 a4
2πL2 −1 − L2
+ 1 + 14 L2
+ L4
that is, there is no term in P −1 . This means that it is no longer the case that the fields are
maximized by choosing P to be arbitrarily small. Instead, the magnetic field is small for
very small charges, and then increases with the charge until it reaches a maximum: only
for still larger charges does it begin to decrease, as shown in figure 5 for a typical choice of
parameters.
In short, for a given value of the angular momentum per unit mass, the maximal
magnetic field can either increase or decrease if the magnetic charge is increased. This is
important when we take into account the condition for superradiance to be avoided, to
which we now turn.
A lengthy but elementary computation shows that P = Pmin always occurs on the
decreasing branch of B + regarded as a function of P. It follows that the maximal magnetic
field that can be attained by a stable extremal AdS4 -Kerr-Newman black hole is given by
– 14 –its value on the equator of the event horizon, when the magnetic charge is precisely Pmin ,
as given in equation (3.3). That is, the maximal possible magnetic field for a given value
of a (taking into account both variations with position and with the magnetic charge, and
therefore denoted by B ++ ) is given by
√
++ 3 (1 + [a/L]) πaL
B = r !. (5.7)
2 2
a2 a 12 2
π`G L2 −1 − L2
+ 1+ L2
+ L2
a2 + (1 + [a/L]) aL
To summarize: the expression in (5.7) is the upper bound on the possible magnetic
JHEP03(2021)068
fields around a stable extremal magnetic AdS4 -Kerr-Newman black hole of given angular
momentum to mass ratio a. With fixed L, it is progressively more stringent for increasing
a. (It is not, however, arbitrarily restrictive: the limiting value of B ++ as a → L is
√
just 1/ ( π`G L).)
The situation regarding the electric field is similar, though slightly more technically
involved.
The electric field also attains is maximum, for given P and a and for points with
r ≥ rH , on the horizon. There, from (2.8), it is given by
−Ξ PrH a cos θ
E(r = rH ) = 2 . (5.8)
2 + a2 cos2 θ
2π rH
As we saw earlier, the electric field on the event horizon attains its maximum either
at the poles, if a is small relative to rH , or at an intermediate value of θ, θ+ ; the maximal
value of the electric field magnitude for given values of a and P is then either
v s
√
u
`2G (1−[a2 /L2 ])2 P 2
2
a2 a2
u
12 2
−3 6 Ξ Pa −1− L2 + 1+ L2 + L2 a +
t
4π
√
E + a/rH ≤ 1/ 3 = s ! 2
`2G (1−[a2 /L2 ])2 P 2
2
5a2 2
π L3 −1 + L2
+ 1+ La 2 + L122 a2 + 4π
(5.9)
or
√
+
√ −9 3 Ξ P
E a/rH ≥ 1/ 3 = s ! ,
`2G (1−[a2 /L2 ])2 P 2
2
a2 a2 12
16πL2 −1 − L2
+ 1+ L2
+ L2
a2 + 4π
(5.10)
where (4.2) has been used.
One can show that, for each given point on the horizon, the electric field behaves,
as the charge is varied, in much the same way as the magnetic field: that is, it increases
for small charges, but decreases for larger charges. As in the magnetic case, the field
decreases over the range of charges permitted by the condition that superradiance be
avoided, so the maximal fields are always attained at the minimal permitted charge, as
given in equation (3.3).
– 15 –√
One can also show that, when P = Pmin , the condition a/rH ≤, ≥ 1/ 3 can be
expressed more directly as a/L ≤, ≥ 1/3. Thus the overall maximal possible electric
fields (that is, for all locations outside the black hole and for all values of the charge
permitted by the need to avoid superradiance) for a given value of a are as follows: first,
if a/L ≤ 1/3, then
v s
√
u
`2G (1−[a2 /L2 ])2 Pmin
2
2 2
a2
u
a 12 2
−3 6 Ξ Pmin a −1− L2 + 1+ L2 + L2 a +
t
4π
E ++ (a/L ≤ 1/3) = s ! 2 ;
2 2 `2G (1−[a2 /L2 ])2 Pmin
2
2
π L3 −1 + 5a 1+ La 2 + L122 a2 +
JHEP03(2021)068
L2
+ 4π
(5.11)
in this case, the maximal field occurs at the poles. Secondly, if a/L ≥ 1/3, the maximal
field is
√
++ −9 3 Ξ Pmin
E (a/L ≥ 1/3) = s ! ;
`2G (1−[a2 /L2 ])2 Pmin
2 2
a2 a2 12
16πL2 −1 − L2
+ 1+ L2
+ L2
a2 + 4π
(5.12)
but now the maximum occurs at θ = θ+ , with a value that may be read off from figure 6.
In both equations, Pmin is to be regarded as a function of a, through equation (3.3). As
in the case of the magnetic field, these bounds become steadily more restrictive for larger
a. Again, however, they are not arbitrarily restrictive: the limiting value of E ++ as a → L
√
is just 1/ (2 π`G L), exactly half of the corresponding quantity in the magnetic case.
In the specific example we have been considering, with a = 50, L = 100 (in Planck
units), we find that the overall maximum possible fields are (in natural rather than Planck
units)
B ++ (a = 50, L = 100) ≈ 0.00598/`2G ; E ++ (a = 50, L = 100) ≈ 0.00389/`2G , (5.13)
where the magnetic field maximum occurs along the equator, and its electric counterpart
occurs at θ ≈ 0.62 (about 36 degrees).
As to the value of `G , it is (in principle) determined holographically through the “dic-
tionary”, which in this case states that `2G = 3L2 /(2Nc )3/2 , where Nc refers to the presence
of this number of coincident M2-branes in the dual field theory [18, 19]. In the present case,
Nc ≈ 483, which is indeed “large”, as usual in discussions of the AdS/CFT correspondence.
Our original observation was that the black hole must rotate in order that the chemical
potential describing the boundary theory should be substantial, as it is for any form of cold
strongly coupled matter. Let us now confirm that this is the case.
6 The chemical potential
We can now compute the chemical potential for matter dual to an extremal magnetic black
hole, with magnetic charge chosen, as above, to maximize the fields subject to the condition
– 16 –JHEP03(2021)068
Figure 6. The value of θ which gives the position, on the event horizon, where the induced electric
field is maximal, for an extremal AdS4 -Kerr-Newman magnetic black hole with L = 100, as a
function of a. (Units are Planck units; bear in mind that, for a/L < 1/3, the maximal field occurs
at θ = 0.)
of stability against superradiance. The result is, from (2.10) and (2.14),
9 1 − [a2 /L2 ] a Pmin
µ = r 2
!; (6.1)
5a2 a2
2πL2 −1 + L2
+ 1+ L2
+ 12
L2
a2 + `2G (1 − [a2 /L2 ])2 Pmin
2 /4π
as above, Pmin is regarded as a function of a, using equation (3.3).
The magnetic field at infinity in this case is, from (2.9) and (2.14),
1 − [a2 /L2 ] Pmin
∞
B = ; (6.2)
4πL2
again, (3.3) allows us to regard this as a function of a.
As explained earlier, we measure the size of the chemical potential by computing
√
the dimensionless quantity µ/ B ∞ . It turns out that this is a monotonically increasing
function of a, which very quickly rises to values well over unity; however, it is bounded,
with limiting value as a → L given as follows:
√
µ 3 L
√ < √ . (6.3)
B∞ 2 π 1/4 `G
– 17 –JHEP03(2021)068
√
Figure 7. µ/ B ∞ as a function of a, for L = 100 (Planck units). The upper bound is ≈ 11.3 in
this case.
For typical values of a, the function is close to its upper bound: see figure 7 for the case
of L = 100 in Planck units. (For the case we have been taking in examples, a = 50, it is
equal to ≈ 10.9.)
√
If µ/ B ∞ is given, then we can use equations (6.1) and (6.2) to compute a (given L
and `G ). We can then compute the maximal possible magnetic and electric fields using
equations (5.7) and either (5.11) or (5.12).
We conclude from (6.3) that, since L must always be substantially larger than `G if the
√
spacetime is to be semi-classical, then µ is indeed reasonably large relative to B ∞ for all
but an extremely narrow range of values of a. In short, absent severe fine-tuning, making
the magnetic extremal black hole rotate does induce an acceptable chemical potential at
infinity.
7 Conclusion
We have used the AdS/CFT duality as a guide to the physics of asymptotically AdS 4 mag-
netically charged black holes. However, we have only used it in the most basic possible way:
all we required was that the boundary field theory should not be obviously unacceptable.
Even this minimal requirement has nevertheless yielded interesting constraints on these
objects. For example, we have learned that they can be stable and that the magnetic and
electric fields around them cannot be arbitrarily prescribed; so that the coronae studied
in [8] may not always be possible in the AdS case.
In fact, the field theory at infinity in this case is somewhat exotic: it is the infrared
fixed point of a maximally supersymmetric SU(N) gauge theory at large N, describing
the low-energy behaviour of M2 branes. Nevertheless the hope has been expressed [42]
– 18 –that such field theories might have much to teach us regarding general features of strongly
coupled CFTs of the kind that appear in studies of critical phenomena. Perhaps, then,
it might be possible to use the AdS/CFT correspondence to deduce further constraints
on magnetic AdS4 black holes, again by insisting that the boundary theory should be
physically reasonable in some specific sense.
Acknowledgments
The author thanks Dr Soon Wanmei for valuable discussions.
JHEP03(2021)068
Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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